Topological quasiparticles and the holographic bulk-edge relation in (2+1)-dimensional string-net models
String-net models allow us to systematically construct and classify (2+1)-dimensional [(2+1)D] topologically ordered states which can have gapped boundaries. We can use a simple ideal string-net wave function, which is described by a set of F-matrices [or more precisely, a unitary fusion category (U...
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| Format: | Article |
| Language: | English |
| Published: |
American Physical Society
2014
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| Online Access: | http://hdl.handle.net/1721.1/89451 https://orcid.org/0000-0002-5874-581X |
| Summary: | String-net models allow us to systematically construct and classify (2+1)-dimensional [(2+1)D] topologically ordered states which can have gapped boundaries. We can use a simple ideal string-net wave function, which is described by a set of F-matrices [or more precisely, a unitary fusion category (UFC)], to study all the universal properties of such a topological order. In this paper, we describe a finite computational method, Q-algebra approach, that allows us to compute the non-Abelian statistics of the topological excitations [or more precisely, the unitary modular tensor category (UMTC)], from the string-net wave function (or the UFC). We discuss several examples, including the topological phases described by twisted gauge theory [i.e., twisted quantum double D[superscript α](G)]. Our result can also be viewed from an angle of holographic bulk-boundary relation. The (1+1)-dimensional [(1+1)D] anomalous topological orders, that can appear as edges of (2+1)D topological states, are classified by UFCs which describe the fusion of quasiparticles in (1+1)D. The (1+1)D anomalous edge topological order uniquely determines the (2+1)D bulk topological order (which are classified by UMTC). Our method allows us to compute this bulk topological order (i.e., the UMTC) from the anomalous edge topological order (i.e., the UFC). |
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